Although Delta appears to be claiming that its website provides a better purchase experience than travel websites, it's more plausible that they're really looking for brand differentiation here in much the same way. Delta likely believes air travel shouldn't be treated as a commodity. For example, it probably thinks it offers a higher quality of service than a discount airline like Sprit.
This attitude isn't uncommon from companies who believe they offer customers a higher quality product than some competitors. The big airlines may be beginning to view themselves like luxury companies. And just like you don't see Gucci or Prada desperate to get on Wal-mart's shelves, these airlines might not be as interested in attracting customers merely looking for the cheapest tickets available through travel websites.
But the story is more complex than that a push for greater brand recognition alone. The USA Today article also explains why Americans is no longer on Orbitz:
American, on the other hand, says that it's not interested in shrinking the number of outside outlets that sell its flights. Instead, it wants Orbitz to switch to a technology system that informs fliers about the ever-growing array of services the airline offers for a fee, such as priority boarding, which bring the airline over $1 billion a year in additional revenue.
At a time when airlines are increasingly relying on fees from a la carte extras as a source of revenue, travel search websites do oversimplify. For example, Spirit charges for carry-on and checked baggage. U.S. Air charges for only checked baggage. On Jet Blue, your carry-on and first checked bags are free. So the ticket prices that pop up on Orbitz for these three airlines may be misleading. Which ultimately provides the cheapest trip depends on more than just the fare shown.
Currently, these sites actually benefit airlines with more fees, because their ticket price appears cheaper, since it doesn't include all the additional charges, initially hidden from view. It's actually a pretty fair argument that airlines want full disclosure about fees so that consumers really know what they're getting. And of course, if the travel site has no way to provide customers with the ability to add on additional fees at the time of booking, like purchasing a premium seat instead, then this could impact airline profits.
But this isn't the only profit-related concern that airlines have about their flights appearing on travel sites. One reason not cited in that USA Today article, but almost certainly pushing airlines away from these sites, has to do with eliminating a middle man. Currently, sites like Orbitz and Expedia rely on global distribution systems to collect the information for their fare searches. Airlines pay a fee fro these services so that travel agents can access their flight pricing data.